Problem 24 · 2014 Math Kangaroo
Hard
Geometry & Measurement
pythagorean-triplesymmetry
In the diagram on the right the following can be seen: a straight line that is the common tangent of two touching circles of radius 1, and a square with one edge on the straight line and the other two vertices one on each of the two circles. How big is the side length of the square?
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Answer: A — \(\dfrac{2}{5}\)
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Hint 1 of 2
By symmetry the square is centred on the touching point; put it on coordinates.
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Hint 2 of 2
Its top corners sit on the circles — plug a corner into a circle's equation.
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Approach: coordinates and one circle equation
- Centres at (±1,1), the tangent line is y = 0; by symmetry the square's base is centred at the origin.
- A top corner (s/2, s) lies on the right circle: (s/2 − 1)² + (s − 1)² = 1.
- This gives 5s² − 12s + 4 = 0, whose sensible (small) root is s = 2/5.
- So the side length is 2/5.
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